The critical two-point function for long-range percolation on the hierarchical lattice

Abstract

We prove up-to-constants bounds on the two-point function (i.e., point-to-point connection probabilities) for critical long-range percolation on the $d$-dimensional hierarchical lattice. More precisely, we prove that if we connect each pair of points $x$ and $y$ by an edge with probability $1-\exp (-\beta \Vert x-y{\Vert }^{-d-\alpha })$, where $0<\alpha <d$ is fixed and $\beta \ge 0$ is a parameter, then the critical two-point function satisfies \[ \mathbb{P}_{β_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+α} \] for every pair of distinct points $x$ and $y$. We deduce in particular that the model has mean-field critical behaviour when $\alpha <d/3$ and does not have mean-field critical behaviour when $\alpha >d/3$.

18 pages